Question

How fast would a \(5.00-\mathrm{g}\) fly have to be traveling to slow a \(1900 .-\mathrm{kg}\) car traveling at \(55.0 \mathrm{mph}\) by \(5.00 \mathrm{mph}\) if the fly hit the car in a totally inelastic head-on collision?

Short answer

Answer: The fly must be traveling at approximately 240,000 m/s before the collision to slow down the car by 5.00 mph.

Step-by-step solution

Identifying the known variables and unknowns

We are given the mass of the car (M = 1900 kg), the mass of the fly (m = 5.00 g = 0.005 kg), the initial speed of the car (v₀(car) = 55.0 mph), and the car's final speed is v₀(car) - 5.00 mph. We need to find the speed of the fly (v₀(fly)) before the collision.

Convert given speeds to m/s

To make our calculations easier, we'll convert the given speeds in mph to m/s. 1 mph = 0.44704 m/s v₀(car) = 55.0 mph × 0.44704 m/s/mph ≈ 24.587 m/s v_f(car) = v₀(car) - 5.00 mph ≈ 24.587 m/s - 2.2352 m/s ≈ 22.352 m/s

Applying the conservation of momentum principle

Since there are no external forces during the collision, the momentum before and after the collision is conserved. We can write the conservation of momentum equation as: momentum_before = momentum_after M × v₀(car) + m × v₀(fly) = (M + m) × v_f(car)

Solve for the unknown v₀(fly)

We have all the values except v₀(fly), so we can now solve for it: v₀(fly) = [(M + m) × v_f(car) - M × v₀(car)]/m Plugging in the values: v₀(fly) = [(1900 kg + 0.005 kg) × 22.352 m/s - 1900 kg × 24.587 m/s]/0.005 kg

Calculate v₀(fly)

Carrying out the calculation: v₀(fly) ≈ 240000 m/s The fly must be traveling at approximately 240,000 m/s to slow the car down by 5.00 mph in a totally inelastic head-on collision.

Key concepts

Conservation of Momentum

The conservation of momentum is a fundamental principle in physics stating that the total momentum of a closed system remains constant if no external forces act upon it. Momentum, a key concept in collision physics, is calculated as the product of an object's mass and velocity.

Let's break it down: in an inelastic collision, two objects collide and stick together after the impact. In such a case, despite the deformation or generation of heat during the collision, the conservation of momentum holds true. For the car and fly problem, since the fly sticks to the car, creating a single system post-collision, we can confidently apply this principle. It implies that the combined momentum of the car and fly before the collision must equal their combined momentum afterwards. This conservation allows us to set up an equation where all the known values can help determine the fly's unknown velocity before the collision.

Momentum Calculation

To calculate momentum, you multiply an object's mass by its velocity. The unit for mass is kilograms (kg) and the unit for velocity is meters per second (m/s). Therefore, momentum has the derived SI unit kg·m/s.

For the exercise at hand, converting units is critical. Speeds were given in miles per hour (mph) but for the consistency of our calculations and adherence to standard SI units, we convert these to meters per second (m/s). After setting up the conservation of momentum equation, we solve for the fly's velocity by rearranging the terms. This step-by-step mathematical process not only gives us the particular value we're looking for but also deepens our understanding of momentum as a product of an object's mass and velocity.

Collision Physics

In the realm of collision physics, an inelastic collision, as our problem showcases, is one in which the colliding objects stick together after impact, which is different from an elastic collision where object's separate post-collision.

The fundamental aspect here is to understand that during an inelastic collision, some of the kinetic energy is not conserved, contrary to momentum. This energy loss is usually converted into other forms of energy, like heat or sound. But since momentum is conserved, we leverage this to solve collision problems in physics. By comprehending this, students can grasp why the seemingly insurmountable speed calculated for the fly is a theoretical representation derived from the conservation of momentum, underlining the vast difference between masses of the two colliding bodies.